A combinatorial identity involving gamma function and Pochhammer symbol

In this present paper, first, we investigate a new generalized Pochhammer's symbol and its various properties in terms of a new symbol $(s; k)$, where $s; k > 0$. Then, we define a new generalization of gamma and beta functions and their various associated properties in the form of $(s; k)$. Also, we define a new generalization of hypergeometric functions and develop differential equations for generalized hypergeometric functions in the form of $(s; k)$. We present that generalized hypergeometric functions are the solution of the said differential equation. Furthermore, some useful results and properties and integral representation related to these generalized Pochhammer's symbol, gamma function, beta function, and hypergeometric functions are presented.

Products of ratios of gamma functions — An application to the distribution of the test statistic for testing the equality of covariance matrices

Let varGamma (x) denote the classical Euler gamma function. The logarithmic derivative psi (x)=[ln varGamma (x)]'=frac{varGamma '(x)}{ varGamma (x)}, psi '(x), and psi ''(x) are, respectively, called the digamma, trigamma, and tetragamma functions. In the paper, the authors survey some results related to the function [psi '(x)]^{2}+ psi ''(x), its q-analogs, its variants, its divided difference forms, several ratios of gamma functions, and so on. These results include the origins, positivity, inequalities, generalizations, completely monotonic degrees, (logarithmically) complete monotonicity, necessary and sufficient conditions, equivalences to inequalities for sums, applications, and the like. Finally, the authors list several remarks and pose several open problems.

This paper introduces some monotonic functions associated with ratios of gamma functions.

Numerous special functions, including the beta function, hypergeometric functions, and other orthogonal polynomials, are closely connected to the gamma function. Recently, gamma function has been enhanced by adding a new parameter. As a consequence, this gamma is called the parametric gamma function or b-gamma function. By utilizing the enhanced gamma function (or the parametric gamma function), we have present a generalization for the special function Rabotnov function. Consequently, new fractal-fractional operators (derivative and integral) involving the generalized Rabotnov function are defined. Analysis is introduced to discover the main properties of the suggested operators. Since the main challenge in the calculus of fractal-fractional operators is to design examples, we illustrate a set of examples including power series. We explore the boundedness of the recommended operators. There are many gains for checking the boundedness of fractal operators in general and fractal-fractional operators in particular. Moreover, as an application, we establish the existence and uniqueness solution of abstract fractal-fractional equation. Examples are presented at the end of the effort. Graphics and computations are observed using MATHEMATICA 13.3 software.•By using the parametric gamma function, the Rabotnov special function is generalized. New fractal-fractional operators are presented using the generalized Rabotnov function with examples;•Boundedness of these operators is investigated, where bounded operators give a strong foundation for understanding linear transformations across normed spaces, with many applications in science and mathematics.•Conditions of the existence and uniqueness of solutions of fractal-fractional differential abstract equation are established.

Exponential functional of a new family of Lévy processes and self-similar continuous state branching processes with immigration

The Elliptic Gamma Function and SL(3, [formula omitted])⋉[formula omitted

The classical theories of shells and curved beams are used to develop the equations of motion of an elastically isotropic spherical shell attached to an elastic equatorial beam of rectangular cross section. The mass densities and elasticities of the shell and beam are, in general, different. Remarkably, for the natural frequencies, the final set of eight linear homogeneous algebraic equations uncouples into two sets of four. The only approximations made are of the same order of magnitude as those inherent in classical shell and beam theory. Although solutions of the shell equations involve Legendre functions (and not polynomials), the final set of algebraic equations involve only trigonometric and gamma functions. Several special exact solutions are given. In Part II, perturbation techniques are used to find the natural frequencies for beam-shell configurations ranging from nearly pure beams to nearly pure shells.

Complete monotonicity of the remainder of the asymptotic series for the ratio of two gamma functions

Considering the equations for some functions involving the first or the second derivatives of the biconfluent Heun function, we construct two expansions of the solutions of the biconfluent Heun equation in terms of incomplete Beta functions. The first series applies single Beta functions as expansion functions, while the second one involves a combination of two Beta functions. The coefficients of expansions obey four- and five-term recurrence relations, respectively. It is shown that the proposed technique is potent to produce series solutions in terms of other special functions. Two examples of such expansions in terms of the incomplete Gamma functions are presented.

A general orientation equation for rigid particles embedded in a continuum has been derived. The equation includes hydrodynamic, thermodynamic, diffusional and kinetic terms, corresponding to physically different orientation mechanisms. Some special solutions of the general equation are presented, and discussed conditions under which the general equation reduces to more simple cases. Numerical examples of orientation in polyethylene subjected to uniaxial forces are discussed.

Multi-adjoint relation equations: Definition, properties and solutions using concept lattices

Published in Petroleum Transactions, AIME, Vol. 207, 1956, pages 281–287. Abstract Rigorous equations for calculating subsurface pressures in flowing and static gas wells, and pressures along horizontal pipelines are presented in this paper. These general equations, based on the mechanical energy balance, contain no assumptions regarding temperature and can be used with either straight-line or curved temperature gradients. The friction factors recommended in this report are based on an absolute roughness of 0.0006 in. Flow is always considered to be turbulent. Although these general flow equations are solved by numerical means, the methods are as convenient as many of those used today to calculate pressures in gas wells and pipelines. These numerical methods are illustrated for flowing and static columns of gas in wells and for flow in pipelines. The authors believe that the use of these methods will result in more accurately calculated pressures in gas wells and pipelines and that the methods are more flexible than any in use at the present time. Also, the methods are easily adaptable to automatic computers. Introduction The problem of calculating subsurface pressures in gas wells and along flowing pipelines has been studied by many investigators with the results that two widely used methods have been presented in technical literature. One approximates conditions in a gas well or pipeline by assuming that temperature and compressibility are constants for the entire gas column.

Chapter 4 - Special functions

We investigate the global behavior of the solutions of rational difference equations. In particular, we study the global stability, the periodic nature, and especially the boundedness character of their solutions. In the first manuscript we study the equation[special characters omitted]and establish that every solution of this equation converges to a (not necessarily prime) period-two solution when γ = β + δ + A and β + A is positive, and that unbounded solutions of this equation exist whenever γ > β + δ + A. In this case, we establish a very large set of initial conditions that produce unbounded solutions. In subsequent manuscripts, we establish conditions under which every solution of the special case of the above equation with α = 0, A = 1, and δ = 1 converges to β + δ. We also present a number of open problems and conjectures related to the general third-order rational difference equation. In all equations, the parameters are nonnegative real numbers and the initial conditions are arbitrary nonnegative real numbers such that the denominators are always positive.